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Number theory Difficulty 3.0 AMC 10/12 Prove it Canada

Find a number between 400 and 420 which occurs in both of the following arithmetic sequences: $17, 22, 27, 32, \ldots$ and $16, 22, 28, 34, \ldots$ . Explain how you got your answer.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.